Biostat/Biomath M257
This lecture introduces GPU computing in Julia.
GPUs are ubiquitous in modern computers. Following are NVIDIA GPUs on today’s typical computer systems.
| NVIDIA GPUs | H100 PCIe | RTX 6000 | RTX 5000 |
|---|---|---|---|
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| Computers | servers, cluster | desktop | laptop |
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| Main usage | scientific computing | daily work, gaming | daily work |
| Memory | 80 GB | 48 GB | 16 GB |
| Memory bandwidth | 2 TB/sec | 960 GB/sec | 576 GB/sec |
| Number of cores | ??? | ??? | ??? |
| Processor clock | ??? GHz | ??? GHz | ??? GHz |
| Peak DP performance | 26 TFLOPS | ??? TFLOPS | ??? TFLOPS |
| Peak SP performance | 51 TFLOPS | 91.1 TFLOPS | 42.6 TFLOPS |
GPUs contain 1000s of processing cores on a single card; several cards can fit in a desktop PC
Each core carries out the same operations in parallel on different input data – single program, multiple data (SPMD) paradigm
Extremely high arithmetic intensity if one can transfer the data onto and results off of the processors quickly
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GPU support by Julia is under active development. Check JuliaGPU for currently available packages.
There are multiple paradigms to program GPU in Julia, depending on the specific hardware.
CUDA is an ecosystem exclusively for Nvidia GPUs. There are extensive CUDA libraries for scientific computing: CuBLAS, CuRAND, CuSparse, CuSolve, CuDNN, …
The CUDA.jl package allows defining arrays on Nvidia GPUs and overloads many common operations.
The AMDGPU.jl package allows defining arrays on AMD GPUs and overloads many common operations.
The Metal.jl package allows defining arrays on Apple Silicon GPU and overloads many common operations.
AppleAccelerate.jl wraps the macOS Accelerate framework, which provides high-performance libraries for linear algebra, signal processing, and image processing on Apple Silicon CPU. This is analog of MKL for Intel CPU.
The oneAPI.jl package allows defining arrays on Intel GPUs and overloads many common operations.
I’ll illustrate using Metal.jl on my MacBook Pro running MacOS Sequoia 15.4. It has Apple M2 chip with 38 GPU cores.
versioninfo()Julia Version 1.11.5
Commit 760b2e5b739 (2025-04-14 06:53 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: macOS (arm64-apple-darwin24.0.0)
CPU: 12 × Apple M2 Max
WORD_SIZE: 64
LLVM: libLLVM-16.0.6 (ORCJIT, apple-m2)
Threads: 8 default, 0 interactive, 4 GC (on 8 virtual cores)
Environment:
JULIA_NUM_THREADS = 8
JULIA_EDITOR = codeLoad packages:
using Pkg
Pkg.activate(pwd())
Pkg.instantiate()
Pkg.status() Activating project at `~/Documents/github.com/ucla-biostat-257/2025spring/slides/09-juliagpu`Status `~/Documents/github.com/ucla-biostat-257/2025spring/slides/09-juliagpu/Project.toml`
[13e28ba4] AppleAccelerate v0.4.0
[6e4b80f9] BenchmarkTools v1.6.0
[bdcacae8] LoopVectorization v0.12.172
[dde4c033] Metal v1.5.1
[37e2e46d] LinearAlgebra v1.11.0using Metal, AppleAccelerate
Metal.versioninfo()macOS 15.4.0, Darwin 24.4.0
Toolchain:
- Julia: 1.11.5
- LLVM: 16.0.6
Julia packages:
- Metal.jl: 1.5.1
- GPUArrays: 11.2.2
- GPUCompiler: 1.3.2
- KernelAbstractions: 0.9.34
- ObjectiveC: 3.4.1
- LLVM: 9.2.0
- LLVMDowngrader_jll: 0.6.0+0
1 device:
- Apple M2 Max (3.000 GiB allocated)using Random
Random.seed!(257)
# generate SP data on CPU
x = rand(Float32, 3, 3)
# transfer data form CPU to GPU
xd = MtlArray(x)3×3 MtlMatrix{Float32, Metal.PrivateStorage}:
0.145793 0.939801 0.479926
0.567772 0.577251 0.81655
0.800538 0.38893 0.914135# generate array on GPU directly
# yd = Metal.ones(3, 3)
yd = MtlArray(ones(Float32, 3, 3))3×3 MtlMatrix{Float32, Metal.PrivateStorage}:
1.0 1.0 1.0
1.0 1.0 1.0
1.0 1.0 1.0# collect data from GPU to CPU
x = collect(xd)3×3 Matrix{Float32}:
0.145793 0.939801 0.479926
0.567772 0.577251 0.81655
0.800538 0.38893 0.914135using BenchmarkTools, LinearAlgebra, Random
Random.seed!(257)
n = 2^14
# on CPU
x = rand(Float32, n, n)
y = rand(Float32, n, n)
z = zeros(Float32, n, n)
# on GPU
xd = MtlArray(x)
yd = MtlArray(y)
zd = MtlArray(z);# SP matrix dot product on CPU: tr(X'Y)
bm_cpu = @benchmark dot($x, $y)BenchmarkTools.Trial: 138 samples with 1 evaluation per sample. Range (min … max): 34.377 ms … 56.968 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 34.923 ms ┊ GC (median): 0.00% Time (mean ± σ): 36.388 ms ± 3.159 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▆██▄▃ ▁ ▁ ███████▅▇▁▅▅▁█▇▇▁▅▇▅▇▇▇▁▅█▁▁▁▇▁▅▅▁▁▅▁▁▁▅▁▅▁▅▁▅▁▁▁▅▁▁▅▁▇▁▅▁▅ ▅ 34.4 ms Histogram: log(frequency) by time 45.5 ms < Memory estimate: 0 bytes, allocs estimate: 0.
# SP matrix dot product on CPU using Apple Accelerate: tr(X'Y)
bm_acc = @benchmark AppleAccelerate.dot($x, $y)BenchmarkTools.Trial: 143 samples with 1 evaluation per sample. Range (min … max): 34.379 ms … 40.565 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 35.034 ms ┊ GC (median): 0.00% Time (mean ± σ): 35.193 ms ± 982.134 μs ┊ GC (mean ± σ): 0.00% ± 0.00% █▂ ▂▅▂ ▃████▆█▇███▅▅▄▃▁▃▁▁▁▁▁▁▃▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃▁▁▁▁▁▁▁▃▃▁▁▁▃▁▁▃ ▃ 34.4 ms Histogram: frequency by time 39.9 ms < Memory estimate: 0 bytes, allocs estimate: 0.
# SP matrix dot product on GPU: tr(X'Y)
# why are there allocations?
bm_gpu = @benchmark Metal.@sync dot($xd, $yd)BenchmarkTools.Trial: 662 samples with 1 evaluation per sample. Range (min … max): 7.447 ms … 8.989 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 7.539 ms ┊ GC (median): 0.00% Time (mean ± σ): 7.556 ms ± 93.986 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▁ ▃█▃▅▁▁▁▃▃ ▁ ▃▄▄▄▇▇█▆███████████▆▆▇█▆▇▇▅█▇▇▆▄▄▄▄▂▄▂▃▂▄▂▃▃▂▂▃▃▂▁▃▁▂▂▁▃▂▂ ▄ 7.45 ms Histogram: frequency by time 7.78 ms < Memory estimate: 21.24 KiB, allocs estimate: 837.
# speedup by Apple Accelerate
median(bm_acc.times) / median(bm_cpu.times)1.0031748038858854# speedup on GPU over CPU
median(bm_cpu.times) / median(bm_gpu.times)4.632175281234197# SP broadcast on CPU: z .= x .* y
bm_cpu = @benchmark $z .= $x .* $yBenchmarkTools.Trial: 142 samples with 1 evaluation per sample. Range (min … max): 34.871 ms … 36.395 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 35.260 ms ┊ GC (median): 0.00% Time (mean ± σ): 35.330 ms ± 307.233 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▄ ▄ ▁██▁▆▄▃▁▄▃▁ ▃ ▁ ▄ ▆█▆█▇▁▆▇▇███████████▇▄▆█▇▇▇▁█▄▆▁▄▄█▄▇▆▄▄▄▄▇▄▄▄▁▁▁▁▁▁▁▁▁▁▁▁▄▄ ▄ 34.9 ms Histogram: frequency by time 36.3 ms < Memory estimate: 0 bytes, allocs estimate: 0.
# SP broadcast on GPU: z .= x .* y
# why is there allocation?
bm_gpu = @benchmark Metal.@sync $zd .= $xd .* $ydBenchmarkTools.Trial: 564 samples with 1 evaluation per sample. Range (min … max): 8.755 ms … 9.588 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 8.855 ms ┊ GC (median): 0.00% Time (mean ± σ): 8.867 ms ± 76.948 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▁ ▁▂ ▃▆▆▂▁█▄▄ ▂▂▂▄▇ ▂ ▄▃▄██▇███████████▇█████▇█▇▆▄▅▇▄▆▄▃▃▃▃▃▂▃▃▂▁▂▁▂▁▂▁▂▂▁▁▁▁▂▁▂ ▄ 8.76 ms Histogram: frequency by time 9.12 ms < Memory estimate: 4.53 KiB, allocs estimate: 177.
# speedup
median(bm_cpu.times) / median(bm_gpu.times)3.9818234247150652# SP matrix multiplication on GPU
bm_gpu = @benchmark Metal.@sync mul!($zd, $xd, $yd)BenchmarkTools.Trial: 6 samples with 1 evaluation per sample. Range (min … max): 913.740 ms … 1.000 s ┊ GC (min … max): 0.00% … 0.00% Time (median): 982.052 ms ┊ GC (median): 0.00% Time (mean ± σ): 963.723 ms ± 37.417 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▁ ▁ █ ▁ ▁ █▁▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█▁█▁▁▁▁▁▁▁▁▁█ ▁ 914 ms Histogram: frequency by time 1 s < Memory estimate: 960 bytes, allocs estimate: 54.
For this problem size on this machine, we see GPU achieves a staggering 9 TFLOPS throughput with single precision!
# SP throughput on GPU
(2n^3) / (minimum(bm_gpu.times) / 1e9)9.626476494557188e12# SP matrix multiplication on CPU
bm_cpu = @benchmark mul!($z, $x, $y)BenchmarkTools.Trial: 2 samples with 1 evaluation per sample. Range (min … max): 3.901 s … 3.905 s ┊ GC (min … max): 0.00% … 0.00% Time (median): 3.903 s ┊ GC (median): 0.00% Time (mean ± σ): 3.903 s ± 2.599 ms ┊ GC (mean ± σ): 0.00% ± 0.00% █ █ █▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁ 3.9 s Histogram: frequency by time 3.9 s < Memory estimate: 0 bytes, allocs estimate: 0.
# SP throughput on CPU
(2n^3) / (minimum(bm_cpu.times) / 1e9)2.2548119538396475e12We see >10x speedup by GPUs in this matrix multiplication example.
# cholesky on Gram matrix
# This one doesn't seem to work on Apple M2 chip yet
xtxd = xd'xd + I
@benchmark Metal.@sync cholesky($(xtxd))ArgumentError: ArgumentError: cannot take the CPU address of a MtlMatrix{Float32, Metal.PrivateStorage}
ArgumentError: cannot take the CPU address of a MtlMatrix{Float32, Metal.PrivateStorage}
Stacktrace:
[1] unsafe_convert(::Type{Ptr{Float32}}, x::MtlMatrix{Float32, Metal.PrivateStorage})
@ Metal ~/.julia/packages/Metal/N2ABH/src/array.jl:266
[2] potrf!(uplo::Char, A::MtlMatrix{Float32, Metal.PrivateStorage})
@ LinearAlgebra.LAPACK ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/lapack.jl:3293
[3] _chol!
@ ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:187 [inlined]
[4] #cholesky!#163
@ ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:268 [inlined]
[5] cholesky!
@ ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:267 [inlined]
[6] cholesky!(A::MtlMatrix{Float32, Metal.PrivateStorage}, ::NoPivot; check::Bool)
@ LinearAlgebra ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:301
[7] cholesky! (repeats 2 times)
@ ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:295 [inlined]
[8] _cholesky
@ ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:411 [inlined]
[9] cholesky (repeats 2 times)
@ ~/.julia/juliaup/julia-1.11.5+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/cholesky.jl:401 [inlined]
[10] macro expansion
@ ~/.julia/packages/Metal/N2ABH/src/utilities.jl:10 [inlined]
[11] var"##core#340"(xtxd#339::MtlMatrix{Float32, Metal.PrivateStorage})
@ Main ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:598
[12] var"##sample#341"(::Tuple{MtlMatrix{Float32, Metal.PrivateStorage}}, __params::BenchmarkTools.Parameters)
@ Main ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:607
[13] _lineartrial(b::BenchmarkTools.Benchmark, p::BenchmarkTools.Parameters; maxevals::Int64, kwargs::@Kwargs{})
@ BenchmarkTools ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:186
[14] _lineartrial(b::BenchmarkTools.Benchmark, p::BenchmarkTools.Parameters)
@ BenchmarkTools ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:181
[15] #invokelatest#2
@ ./essentials.jl:1055 [inlined]
[16] invokelatest
@ ./essentials.jl:1052 [inlined]
[17] #lineartrial#46
@ ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:51 [inlined]
[18] lineartrial
@ ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:50 [inlined]
[19] tune!(b::BenchmarkTools.Benchmark, p::BenchmarkTools.Parameters; progressid::Nothing, nleaves::Float64, ndone::Float64, verbose::Bool, pad::String, kwargs::@Kwargs{})
@ BenchmarkTools ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:299
[20] tune!
@ ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:288 [inlined]
[21] tune!(b::BenchmarkTools.Benchmark)
@ BenchmarkTools ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:288
[22] top-level scope
@ ~/.julia/packages/BenchmarkTools/1i1mY/src/execution.jl:461xtx = collect(xtxd)
@benchmark LinearAlgebra.cholesky($(Symmetric(xtx)))BenchmarkTools.Trial: 1 sample with 1 evaluation per sample. Single result which took 7.649 s (0.00% GC) to evaluate, with a memory estimate of 1.00 GiB, over 3 allocations.
@benchmark AppleAccelerate.cholesky($(Symmetric(xtx)))BenchmarkTools.Trial: 1 sample with 1 evaluation per sample. Single result which took 7.635 s (0.00% GC) to evaluate, with a memory estimate of 1.00 GiB, over 3 allocations.
We don’t see GPU speedup of Cholesky at the moment.
# elementwise function on GPU arrays
fill!(yd, 1)
bm_gpu = @benchmark Metal.@sync $zd .= log.($yd .+ sin.($xd))
bm_gpuBenchmarkTools.Trial: 563 samples with 1 evaluation per sample. Range (min … max): 8.769 ms … 9.569 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 8.871 ms ┊ GC (median): 0.00% Time (mean ± σ): 8.879 ms ± 67.372 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▁▁▁ ▂▂▃█▄▆▁▄▂▃▆▁▃▂▄▃ ▂▃▅▆▆▆███▇█████████████████▇▅▆▇▇▄▄▄▅▂▄▄▄▃▁▃▁▂▃▁▁▁▂▂▂▂▁▁▁▁▂ ▄ 8.77 ms Histogram: frequency by time 9.1 ms < Memory estimate: 4.53 KiB, allocs estimate: 177.
# elementwise function on CPU arrays
x, y, z = collect(xd), collect(yd), collect(zd)
bm_cpu = @benchmark $z .= log.($y .+ sin.($x))
bm_cpuBenchmarkTools.Trial: 2 samples with 1 evaluation per sample. Range (min … max): 2.750 s … 2.755 s ┊ GC (min … max): 0.00% … 0.00% Time (median): 2.753 s ┊ GC (median): 0.00% Time (mean ± σ): 2.753 s ± 3.053 ms ┊ GC (mean ± σ): 0.00% ± 0.00% █ █ █▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁ 2.75 s Histogram: frequency by time 2.75 s < Memory estimate: 0 bytes, allocs estimate: 0.
# Speed up
median(bm_cpu.times) / median(bm_gpu.times)310.27966704755283GPU brings great speedup (>50x) to the massive evaluation of elementary math functions.
bm_cpu = @benchmark z .= tanh.($x) # on CPU
bm_cpuBenchmarkTools.Trial: 4 samples with 1 evaluation per sample. Range (min … max): 1.600 s … 1.608 s ┊ GC (min … max): 0.00% … 0.00% Time (median): 1.605 s ┊ GC (median): 0.00% Time (mean ± σ): 1.604 s ± 3.526 ms ┊ GC (mean ± σ): 0.00% ± 0.00% █ █ █ █ █▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁█ ▁ 1.6 s Histogram: frequency by time 1.61 s < Memory estimate: 16 bytes, allocs estimate: 1.
bm_acc = @benchmark z .= AppleAccelerate.tanh.($x) # AppleAccelerate
bm_accBenchmarkTools.Trial: 27 samples with 1 evaluation per sample. Range (min … max): 190.344 ms … 192.504 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 191.780 ms ┊ GC (median): 0.00% Time (mean ± σ): 191.633 ms ± 508.052 μs ┊ GC (mean ± σ): 0.00% ± 0.00% █ ▃ █ █ ▃ ▇▁▁▁▁▁▁▇▁▁▁▇▇▁▁▁▁▁▁▁▁▁▇▁▁▁▁▁▁▁▁█▁▁▇▁▁█▁▇█▇█▁▇▇▇▇▁▇█▁▁▁▁▁▁▁▁▁▇ ▁ 190 ms Histogram: frequency by time 193 ms < Memory estimate: 16 bytes, allocs estimate: 1.
AppleAccelerate.jl accelerates the evaluation of tanh function by
median(bm_cpu.times) / median(bm_acc.times)8.370156534894807bm_mtl = @benchmark zd .= Metal.@sync tanh.($xd) # Metal
bm_mtlBenchmarkTools.Trial: 74 samples with 1 evaluation per sample. Range (min … max): 42.764 ms … 3.338 s ┊ GC (min … max): 0.00% … 0.00% Time (median): 47.531 ms ┊ GC (median): 0.00% Time (mean ± σ): 99.056 ms ± 384.921 ms ┊ GC (mean ± σ): 0.00% ± 0.00% █▃ ██▆▄▄▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▄ ▁ 42.8 ms Histogram: log(frequency) by time 477 ms < Memory estimate: 8.93 KiB, allocs estimate: 335.
Metal.jl accelerates the evaluation of tanh function by
median(bm_cpu.times) / median(bm_mtl.times)33.77199471480576